problems-8-223-F07

problems-8-223-F07 - M ATH 223 Reading 2.10 3.13.3 P ROBLEM...

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Unformatted text preview: M ATH 223 Reading. 2.10, 3.13.3. P ROBLEM S ET 8 D UE : 30 October 2007 Problems from the book: the starred problems may be carefully graded. You will be given a "completeness" score, depending on how carefully you complete these problems. 2.10: 1abc, 2, 3, 8, 9, 13, 16 3.1: 1, 2, 8 , 9, 19 3.2: 1, 4 , 5, 6 Additional problems: These will be carefully graded! 1. Let f : R2 R2 be defined by f(x, y) = (x2 - y2, 2xy). a. Show that f is one-to-one on the set A consisting of all (x, y) with x > 0. [H INT: If f(x, y) = f(a, b), then |f(x, y)| = |f(a, b)|.] b. What is the image B = f(A)? 0 . c. If g is the inverse function, find Dg 1 2. Let f : Rn Rn be given by the equation f(x) = |x|2 x. Show that f is differentiable - and that f carries the unit ball centered at 0 to itself in a one-to-one fashion. Show, - however, that the inverse function is not differentiable at 0 . ...
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